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May 17, 2008

Convenient Categories of Smooth Spaces

Posted by John Baez

Ever since Urs and I first started working on higher gauge theory, we’ve needed something more general than manifolds. You’ve probably heard about the misanthrope who loves humanity as a whole but can’t stand anyone individually. It’s the other way with manifolds. They’re very nice individually — but the category of manifolds as a whole is really annoying. It lacks almost all the properties we expect from a good category!

Grothendieck faced a similar problem long ago in algebraic geometry. He realized that a nice category that includes nasty objects is better than a nasty category with only nice objects. This is why he generalized algebraic varieties and invented ‘schemes’. You can do lots of constructions with schemes that you can’t do with varieties. Sometimes these constructions will lead to nasty schemes. But, that’s a worthwhile price to pay.

It’s time to do the same thing in differential geometry! So, that’s what my grad student Alex Hoffnung has been pondering:

Abstract:
A ‘Chen space’ is a set XX equipped with a collection of
‘plots’ — maps from convex subsets of Euclidean space
into XX — satisfying three simple axioms.
In many respects Chen spaces provide a more convenient
setting for differential geometry than the category of smooth
manifolds. For example, any subspace or quotient space
of a Chen space is a Chen space, the space of smooth maps
between Chen spaces is a Chen space, and the category of
Chen spaces has all limits and colimits. Souriau’s ‘diffeological
spaces’ share all these properties. Here we give a unified treatment
of both formalisms. Following ideas of Dubuc, we show that Chen
spaces, diffeological spaces, and even simplicial complexes are
examples of ‘concrete sheaves on a concrete site’. As a result,
they are locally cartesian closed categories with all limits
and colimits, and a weak subobject classifier. For the benefit
of differential geometers, our treatment explains much of the
category theory that we use.

Here’s one way the category of smooth manifolds is annoying: the space of all smooth maps between smooth manifolds is not a smooth manifold! Sure, it’s some sort of infinite-dimensional manifold, but then the space of maps between those becomes even more tricky to think about. It can be all be worked out, and Kriegl and Michor did it as well as anyone could… but one can’t help but wondering if there’s an easier approach.

Another problem, still more devastating, is that if you have two smooth maps between smooth manifolds

f,g:X→Yf, g : X \to Y

the set

{x∈X|f(x)=g(x)} \{ x \in X | f(x) = g(x) \}

is not usually a manifold. More precisely, the category of manifolds doesn’t have equalizers. It also doesn’t have coequalizers.

In the 1970s Chen invented a category of spaces that avoids all these problems! So does the category of diffeological spaces, invented by Souriau in 1980. These are just two of many concepts of ‘smooth space’ — for a review, see Andrew Stacey’s paper Comparative smootheology. But, these two are similar enough that we can study them in a unified way. They are both categories of ‘concrete sheaves on a concrete site’. So, that’s how we tackled them… with a lot of help from various café regulars, and also James Dolan, who taught us some of the necessary topos theory.

Topos theory? Yes: these categories of smooth spaces aren’t topoi — but they’re pretty close: they’re ‘quasitopoi’. That sounds unpleasant: any concept whose name starts with ‘quasi’ or ‘pseudo’ sounds a bit defective and deranged. But quasitopoi are actually quite nice, and our paper will gently teach you about them, a bit at a time.

Our paper isn’t done yet. The later sections, in particular, still need a lot of polishing. But only the diehards will read those sections anyway: that’s where all the yucky proofs are. The rest is pretty close to done — close enough to show you! We’d really appreciate your comments, corrections, suggestions, questions, and so on.

Posted at May 17, 2008 3:44 AM UTC

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Re: Convenient Categories of Smooth Spaces

p. 4, 4th full paragraph: one of “definitions notions” probably should not be there

p. 6: the itemize list is referred to by numbers of items. Probably you want to make it enumerate

p.6-7: concerning exposition, why not discuss diffeological spaces first, then Chen spaces? The diffeological spaces convey the idea and the problem with convex subsets is then clearly seen to be a technical subtlety that is not the heart of the matter.

p. 7, last item (6): is the embedding
Top↪C∞
Top \hookrightarrow C^\infty
full and faithful? How does that functor combine with the one C∞→TopC^\infty \to Top on p. 8?

p. 9, first full paragraph: “functor that is the identity on maps” = full and faithful (?)

Re: Convenient Categories of Smooth Spaces

Thanks for all these corrections and comments, Urs! This blog entry of mine is one of those “please leave the room and go read a book I just wrote” kind of things — not optimized for starting a fun conversation. So, I’m glad you left the room, read the book, and found your way back here with something to say!

is the embedding

Top↪C∞ Top \hookrightarrow C^\infty

full and faithful?

Let’s do the case of diffeological spaces. I’ll present the problem in a self-contained form so other folks can jump in and tackle it.

Given a topological space XX, we define a plot in XX to be a continuous map φ:D→X\varphi : D \to X where DD is an open subset of some ℝn\mathbb{R}^n.

Then, we define a function f:X→Yf : X \to Y between topological spaces to be smooth if, when we compose it with any plot in XX, we get a plot in YY.

Every continuous map f:X→Yf : X \to Y is smooth in this sense. So, we get a functor Top→C∞Top \to C^\infty. This functor is clearly faithful: different continuous maps give different smooth maps. But is it full? That’s the tricky part. I doubt it’s true.

But let me pose this puzzle in a completely self-contained way, so people can try it:

It seems unlikely, because we’re only probing XX by continuous maps from open subsets of Euclidean space, and there are nasty topological spaces that have very few such maps.

Okay, I guess I can settle the question myself now: the answer to the puzzle is no. I’ll let other people take a crack at finding counterexamples!

So, Urs: both for diffeological spaces and Chen spaces, this functor is faithful but not full. So, I wouldn’t call it an ‘embedding’. I’m never sure what an ‘embedding’ of categories is supposed to mean — my superiors never told me — but I like to reserve that term for a full and faithful functor.

How does that functor combine with the one C∞→TopC^\infty \to \Top on p. 8?

I’m getting a little tired of thinking about this now, but the main thing to say is that the functor Top→C∞Top \to C^\infty is a sort of goofy functor that mainly seems useful for creating ridiculous smooth spaces that are counterexamples to naive conjectures. That’s why I didn’t say much about it. On the other hand, the faithful but not full functor C∞→TopC^\infty \to \Top seems to be useful.

Re: Convenient Categories of Smooth Spaces

I guess you’re right, there’s no good reason to do Chen first — except for his chronological precedence, which fits into our historical narrative. But that could be reworked.

The paper started out being mainly about Chen spaces, with diffeological spaces as a footnote, but then we started giving diffeological spaces almost equal prominence, to increase our customer base and emphasize the need for a unifying concept of ‘concrete sheaf on a concrete site’.

By the way — back when I said our concept of ‘concrete sheaf’ was a bit than your notion of ‘quasirepresentable sheaf’, I didn’t mean to imply it covered all the cases of your idea! It’s not a complete substitute.

It’s just very elegant in the cases where it works, because a concrete site is just a site with extra properties — and then the concrete sheaves on this are just sheaves with extra properties…. whereas your ‘quasirepresentability’ involves extra structure on the site category: a chosen functor to SetSet.

As category theorists love to emphasize, extra properties have a tendency to take of themselves, while extra structure requires constant maintenance.

The classic example is ‘categories with finite products’ versus ‘symmetric monoidal categories’. The former notion is a property of categories: you can just take a category and check to see if it has finite products. The latter is an extra structure: you can take a category and equip with the structure of a symmetric monoidal category in various ways. This distinction winds up making the former notion more powerful: there are lots more theorems about categories with finite products…

… but, when nature gives us a symmetric monoidal category that’s not a category with finite products, we must accept this fact and learn to live with it! There’s no point in complaining.

The same is likely to be true about quasirepresentable sheaves. More nice theorems will be true about concrete sheaves (see our paper)… but when nature hands us a quasirepresentable sheaf of a more general sort, we should not disdain it.

Re: Convenient Categories of Smooth Spaces

p. 18: concerning the terminology “sections” versus “plots” for the set X(D)X(D) assigned by a sheaf to a domain DD, I think one wants to think “sections” iff the site is one of (open) subsets of some fixed space, while for sites such as those of all open subsets in the world, one thinks “plot”.

There must be a good abstract nonsense way to think of this difference, which I wish I knew. It’s a relevant source of confusing in the context of comparative smootheology: for instance when talking about locally ringed spaces, these are conceived in terms of sheaves on open subsets of a fixed underlying (topological) space. While a sheaf, this is morally the thing of out-maps, of course, and plays an opposite (dual, maybe) role to the kind of sheaves of in-maps that arise in Chen spaces and in diffeology.

p. 19, last sentence of first paragraph: probably one superfluous “underlying”

Re: Convenient Categories of Smooth Spaces

Urs wrote:

p. 18: concerning the terminology “sections” versus “plots” for the set X(D) assigned by a sheaf to a domain D, I think one wants to think “sections” iff the site is one of (open) subsets of some fixed space, while for sites such as those of all open subsets in the world, one thinks “plot”.

There must be a good abstract nonsense way to think of this difference, which I wish I knew. It’s a relevant source of confusion in the context of comparative smootheology…

Yes, this is an interesting issue — at the very least psychologically, but perhaps mathematically as well.

It could be related to another distinction that used to confuse the heck out of me: the distinction between a ‘petit topos’ and a ‘gros topos’.

I’ve still never seen these terms defined, but my impression is that a typical ‘petit topos’ is the topos of sheaves on a space, while a typical ‘gros topos’ is the topos of sheaves on some category of spaces.

So, maybe we should say ‘section’ when dealing with a petit topos, and ‘plot’ when dealing with a gros topos.

But wait… I’m wrong about the ‘gros topos’! Mac Lane and Moerdijk say:

In the French school, one calls Sh(X)Sh(X) the petit (small) topos associated with the space X, and Sh(T)/y(X)Sh(T)/y(X) the gros (large) topos associated with the same space. Thus the small and large topoi associated with XX are homotopy equivalent.

Here TT is the category of topological spaces with its usual Grothendieck topology, X∈TX \in T is a fixed space, Sh(T)Sh(T) is the category of sheaves on TT, y(T)∈Sh(T)y(T) \in Sh(T) is the representable sheaf corresponding to XX, and Sh(T)/y(T)Sh(T)/y(T) is the ‘slice category’ consisting of sheaves on TT equipped with a map to y(T)y(T).

So, the point is that the ‘gros topos’ Sh(T)/y(T)Sh(T)/y(T) is not really very different from the ‘petit topos’ Sh(T)Sh(T). I guess I’ll have to call Sh(T)Sh(T) an ‘grand gros topos’ to indicate how much bigger it is.

I was very confused when first learning topos theory in an unsystematic hit-or-miss way, because there seemed to be many flavors: ‘topos’, ‘elementary topos’, ‘Grothendieck topos’, ‘petit topos’, ‘gros topos’. The real problem, it turned out, was that the same word means different things in different periods of history. Grothendieck invented his sort of ‘topos’ — the category of sheaves on a site. Then Lawvere and Tierney invented the ‘elementary topos’,
which is just a category with extra properties. Then everyone decided that the ‘elementary topos’ was more fundamental, called that a ‘topos’, and invented the term ‘Grothendieck topos’ for Grothendieck’s original invention!

‘Petit’ and ‘gros’ are just extra frills, explained above.

Anyway, back to your original point: perhaps we should think ‘section’ when referring to sheaves on a mere poset, e.g. the poset of open subsets of a space… and think ‘plot’ when referring to sheaves on a full-fledged category.

Re: Convenient Categories of Smooth Spaces

A while back John made some useful comments about the distinction – or not – between petit and gros topoi.

I had a little bit of discussion about this (but not enough, lunch break is always too short) with Ieke Moerdijk in Lausanne # from which I got the impression that it is right to address a category of sheaves on a site such as (all) CartesianSpacesCartesianSpaces as a gros topos.

Here is a genuine technical question I have about this (and should have asked before):

it seems that lots of the usual technology about sheaves on sites of open subsets of a given topological space generalizes straightforwardly to such gros toposes.

For instance:

A morphism of sheaves is epi/mono/iso if and only if this is true for the induced morphisms on every stalk.

This is true for sheaves on open subsets of a given topological space, with the usual definition of stalk. There is now an obvious generalization of the notion of stalk to sheaves on arbitrary concrete sites. And an obvious generalization of the above statement.

Is the statement then still correct? What is a good reference for this?

Re: Convenient Categories of Smooth Spaces

This response is a bit off the cuff, but the question seems closely related to the one where we ask: which sheaf categories have “enough points”? Let me explain:

A “point” of a topos EE is a geometric morphism f*:Set→Ef_{*}: Set \to E. Now a geometric morphism is the right adjoint part of an adjoint pair, where the left adjoint

f*:E→Setf^{*}: E \to Set

is (by definition of geometric morphism) required to be “left exact”, that is, preserve finite limits. In the case where EE is the category of sheaves on a space, each “point” comes from a point of the space, and the left adjoint parts are the functors which take stalks at those points.

Expanding the phrase “stalk functor” to apply to any left exact left adjoint

f*:E→Setf^*: E \to Set

for any topos EE, it’s always true that if h:X→Yh: X \to Y is epi/mono in EE, then the corresponding map f*(h)f^*(h) is epi/mono in SetSet. But for some toposes (or for some sites used to generate toposes), there may not be enough such stalk functors (that is, not enough “points”) around to be able to turn the statement around, to say that hh is epi/mono if its image under every stalk functor is epi/mono.

Let me draw your attention though to the discussion in the topos theory book by Mac Lane and Moerdijk, chapter IX, section 11, where they discuss a famous theorem called Deligne’s theorem: given a site whose underlying category admits all finite limits and where the covering families are finite families, the category of sheaves has enough points. Theorems like this may be what you’re after.

Then I thought the stalk of XX at xx is
Xx:=colimVx∈Neighbourhoods(x)X¯(Vx).
X_x := colim_{V_x \in Neighbourhoods(x)} \bar X(V_x)
\,.

I hope that for such concrete sites SS something like this is equivalent to left exact left adjoint functors f*:E→Setf^* : E \to Set, because otherwise I’ll have to worry about being able to actually use this notion of stalk in applications…

Re: Convenient Categories of Smooth Spaces

historically, the first appearance of a gros topos in the literature
is a topological gros topos by Grothendieck and Verdier in SGA IV,
Exposé VI (on page 316).

Gros toposes are those which are “category-of-space like”, opposed
to petit toposes that are “space like”.

The distinction between these two types of toposes can be best seen
on the example of a topological space XX. The petit (small) topos
associated to XX is familiar (Grothendieck) topos Sh(X)Sh(X) of sheaves on
XX. The gros (large) topos associated to the same space XX is the
slice topos Sh(Top)/y(X)Sh(Top)/y(X) where TopTop is a gros site of topological
spaces together with a natural Grothendieck (pre)topology whose
covering families are usual coverings of topological spaces, and
y(X)y(X) is a representable sheaf.

On the more general grounds, the distinction between the two types
of toposes was first systematically investigated by Lawvere, in his
two papers

Re: Convenient Categories of Smooth Spaces

t8m8r wrote:

It seems that the paper is not accessible by public.

I thank you and Daniel for pointing out this problem, and Urs for cleverly getting around it! I’ve fixed the problem and taken the liberty of deleting your comments (since now they’ll only confuse the poor readers of this blog). Please don’t take offense.

Why, on UNIX, when I copy a file with permissions set to 744 (readable by the whole world), do the permissions get reset to 700 (not readable by the whole world)?

I guess it’s a way to prevent people from accidentally exposing secret information to the world at large… but for me it only means I have to carefully remember to reset the permissions to 744 — which I often forget, on this blog.

Re: Convenient Categories of Smooth Spaces

This generally happens when your “umask” is set to keep people from reading things. And your sysadmins often set it to keep private things private. You probably want a umask of 0022, which will prevent others from writing newly created files, but not from reading them. Any reasonable unix shell has a way of setting this, usually with a builtin command called “umask”.

Re: Convenient Categories of Smooth Spaces

Re: Convenient Categories of Smooth Spaces

Check your umask setting. By default files are copied with the same permissions bits, but if you have a umask set it is bit-wise and-ed with the permissions bits of the file being copied to get the new permission bits. It also controls the default permission bits on newly created files as well. The umask is typically set in the configuration file for what ever command shell you are using. Some systems have a default umask that is chosen by the system administrators and you have to override it in the shell config file, if you want something different. Your umask is likely set to 0700 and you probably what it set to 0744 instead or even 0755 if you want to preserve execute bits on copied files too.

Re: Convenient Categories of Smooth Spaces

No, no, no, no! Do not change your umask setting – at least, not to a more permissive setting (you may want to make it more restrictive). A good sysadmin will set a restrictive default umask for a Good Reason. Local users (or lusers as they’re known in the trade) will not change it because they don’t know how to, and more advanced users will not change it because they know that they shouldn’t.

Here’s a cautionary tale for you. When I was a graduate student, a friend asked me for advice on some configuration for some program or other. Rather than walk around the lab, I had a look at the files in his directory and told him what to do. He was a little surprised that I could do this, so I showed him that I could also read his email and his thesis.

Only have a permissive umask if you trust everyone who has access to your computer (except the sysadmin, of course; they read your email anyway). Including the cleaner.

A better solution is to use the -p flag to the cp command. That says “Copy the file with all its permissions intact”. If this is what you always want to do then you can create an alias to it so that when you invoke cp then it really calls cp -p. How you do that depends on what shell you use, and what your current set-up is. I shan’t list all the possibilities here but if you want help on this then feel free to email me (or if you prefer, you could send me your password and I’ll set it up for you). Of course, you don’t have to set the alias to cp. You could use cpp for “CoPy with Permissions” (yes, I do know what I’ve just said before anyone flames me).

To answer John’s other question on this, on most UNIX systems, umask is a once-for-all setting. There are some filesystems that allow a little more fine tuning, such as the aptly named “Andrew File System”, and also if you set the permission of the top directory correctly then you don’t need to worry about the permissions of the files it contains (you need read access to the directory before you can read any of its contents – essentially, if you lock the house then you don’t need to worry about locking the study. Unless your kids are in the habit of leaving the back door open.).

It’s also worth doing a check every now and then because not all programs respect the umask. I’ve just checked and it appears that my web-browser doesn’t do this. About 5% of my files and directories are world-readable, mostly downloaded stuff.

Re: Convenient Categories of Smooth Spaces

Andrew wrote:

A better solution is to use the -p flag to the cp command. That says “Copy the file with all its permissions intact”. If this is what you always want to do then you can create an alias to it so that when you invoke cp then it really calls cp -p.

That sounds like a better idea, and I can do it.

I already made up a ‘put’ command, which I use to put files in the directory for my website while simultaneously adjusting the permissions so everyone can see them. I use this command at least 5 times a day…

… but occasionally I have a file already in my website, and I copy it to another location in my website, and I forget that the permissions get reset to make the copy unreadable by everyone in the world. And this is annoying.

I hope to get some comments from you on this smooth spaces paper, but UNIX help is also great.

Btw, do you plan to include in your paper a proof that ChenChen and SouriauSouriau are not equivalent? Right now our paper optimistically refers to yours for a proof of that.

(The -exec bit changes the permissions to the right ones; if you just want to find the files and directories with the wrong permissions leave off the -exec and everything afterwards. Then you can decide manually whether or not to change things).

It would be technically possible to write a shell script that detected whether or not you were in your web directory and ensured that the permissions were correct, but I think that cp -p combined with find is the simplest (and therefore least likely to go wrong!).

I shall comment on the paper when I’ve read it through carefully - if there hadn’t been a UNIX point to make then I’d probably have put a short comment along the lines of “great to see that this is nearing completion, I shall read it carefully and comment when I can, but I’ve had my Algebraic Topology hat on this last week and have my Norwegian exam on Thursday so might be a little slower than hoped for.”

However, since you ask a specific question I shall give a specific answer:

Btw, do you plan to include in your paper a proof that Chen and Souriau are not equivalent? Right now our paper optimistically refers to yours for a proof of that.

I do; or, since my exam is incipient, ja, det gjør jeg. I’m in the middle of rewriting my paper at the moment and shall include the proof of the above result (always assuming I have a correct proof!). Indeed, one of my reasons for carefully reading your paper is to see what I need to include in mine!

Re: Convenient Categories of Smooth Spaces

I just fixed all the problems Urs has noted so far. Thanks, Urs!

Urs wrote:

p. 9, first full paragraph: “functor that is the identity on maps” = full and faithful (?)

No, these concepts are very different. Fullness and faithfulness are useful properties of functors, invariant under natural isomorphism. ‘Acting as the identity on morphisms’, on the other hand, is a crazy, evil property:

F(f)=f F(f) = f

It just so happens to be the quickest way to describe what the functor we’re talking about does to morphisms: it leaves them alone!

We’re setting up a functor from Chen spaces to diffeological spaces. So, we
need to turn a smooth map between Chen spaces into a smooth map between diffeological spaces. And, we’re doing this by not changing this map at all! Of course this only makes sense because this map is a function between sets — and we’re not changing those sets, either.

I agree that this is a bit confusing to someone who has fully incorporated category theory into their thinking. It’s an old fashioned, set theoretic thing to say.

Anyway, F(f)=fF(f) = f implies the functor FF is faithful — but not necessarily full, since there could be new smooth maps between diffeological spaces that didn’t come from maps between Chen spaces.

The point of this section is to review Andrew Stacey’s work on these functors. I want it to be an easy place to find everything you might ever need to know about the relation between Chen spaces and diffeological spaces! It’s a bit technical, but it’s the sort of thing we comparative smootheologists wonder about.

Re: Convenient Categories of Smooth Spaces

Here’s a question I’ve asked before around here but which I don’t think has been answered:

Are any of these kinds of smooth spaces the result of gluing certain local models together along certain equivalence relations?

In other words, is there a category MM (=”local models”) with a Grothendieck topology such that our category SS of spaces is equivalent to the full subcategory of sheaves on MM containing MM and closed under quotients by equivalence relations of a certain type TT. I’m not exactly sure what the key axiomatic properties of TT should be, but if it’s imporant, I could probably take a stab at it. Maybe I should add that MM should be simpler in some sense than our category CC of smooth spaces. Otherwise, you could probably just take MM to be CC itself!

For example, the category of schemes satisfies this property. So does the category of algebraic spaces.

Re: Convenient Categories of Smooth Spaces

This doesn’t precisely answer your question (which will take more thought), but in Proposition 49 near the very end of our paper we show — both for Chen spaces and diffeological spaces — that every such space is a colimit of very nice ones, namely the ‘domains of plots’. For Chen spaces, these domains are just convex subsets of ℝn\mathbb{R}^n’s. For diffeological spaces, they’re open subsets of ℝn\mathbb{R}^n’s.

Also, in both cases, the domains form a category with a Grothendieck topology such that the smooth spaces (either Chen or diffeological) are precisely the ‘concrete’ sheaves on this category. The smooth spaces form a full subcategory of the category of all sheaves.

So, I know some stuff that sounds vaguely similar to what you’re asking about, but I didn’t actually answer your question.

Re: Convenient Categories of Smooth Spaces

Thanks. I ask because it seems to me that to prove non-trivial results, you’d need to reduce things to the local case and then glue. (Though maybe this says more about me than about the mathematics…) So if XX is a space of interest, you typically prove something about it by finding a simpler space UU covering it. A common example is where UU a disjoint union of local models covering XX. Then U×XUU\times_X U is an equivalence relation on UU. Because UU is simple, you’ll probably be able to prove your theorem about it. If U×XUU\times_X U is also very simple, for example it might also be a disjoint union of local models, then you’ll probably be able to prove your theorem about it, too. Finally, if the equivalence relation U×XUU\times_X U on UU is sufficiently nice, then you’ll be able to prove your theorem about its quotient—in other words, XX.

As I said, that’s pretty much the only way I can think of of proving results that aren’t purely category-theoretic. So, while I would agree that you want to have a nice category rather than a category of only nice objects, I would add that you want your category to be the minimal one with certain niceness properties, otherwise it will contain tons of stuff that isn’t really accessible. One reasonable way of doing this would be to take the closure of your category of local models under quotients by equivalence relations with very nice properties. This gives a nice ‘geometric’ category of spaces inside a topos. The topos is also very nice (being nice is the whole point of toposes!), but it isn’t very geometric. So if you can take all colimits in your category, I would be worried that it would be too big.

I might be too attached to this way of building categories of spaces. So I’d also be happy if someone could give me a really convincing example where it isn’t what we want!

Re: Convenient Categories of Smooth Spaces

One more thing. In theorem 9, you say that you have all small colimits in your category. Of course, any sheaf is a *large* limit of representable sheaves. So is it accurate to say that the way that your category fails to be the category of all sheaves is purely set-theoretic?!?

Am I missing something? (I think I’m assuming that colimits in your category agree with those in the ambient topos. Is this the problem? Also, apologies if this is discussed in the paper. I haven’t actually read it yet!)

Re: Convenient Categories of Smooth Spaces

James wrote:

So is it accurate to say that the way that your category fails to be the category of all sheaves is purely set-theoretic?!?

No, the concrete sheaves are really very different from the arbitrary sheaves — not just in subtle ways involving size issues, or nuances of set theory. Check out definitions 16 and 17 around page 18 of the current draft: that’s where the paper really starts.

Also, apologies if this is discussed in the paper. I haven’t actually read it yet!

No problem. A paper is at best a poor substitute for its author, and I’m blogging about this paper because I want to talk about it. But it’s my bedtime now, so while I snooze you might want to peek at the final subsection, where we discuss colimits of concrete sheaves.

Re: Convenient Categories of Smooth Spaces

I would guess that colimits in the ambient topos do not gnerally agree with colimits in your category, only their concretizations do. In fact, now that it’s the morning, I can say that the right attitude would have been for me to be surprised if it were otherwise!

So I take back my previous comment. But the one before that I stand by (for now).

Re: Convenient Categories of Smooth Spaces

I was at the Mathematical School, where the Master taught his Pupils after a Method scarce imaginable to us in Europe. The Proposition and Demonstration were fairly written on a thin Wafer, with ink composed of a Cephalick Tincture. This the Student was to swallow upon a fasting Stomach, and for three Days following eat nothing but Bread and Water. As the Wafer digested, the Tincture mounted to his Brain, bearing the Proposition along with it.

James wrote:

So I take back my previous comment. But the one before that I stand by (for now).

Okay. Let me say a bit more about that…

There’s a whole book of theorems about diffeological spaces available free online, by Iglesias–Zemmour. The usual method is to work ‘one plot at a time’ and then show that all the constructions behave well under change of plot. You may or may not consider this to be ‘reducing things to the local case and then gluing’… but if you take a sufficiently broad-minded attitude, I think you might.

Re: Convenient Categories of Smooth Spaces

Thanks, John! I have a few more questions, if you’re up for it.

1. As I said before, the way general spaces are constructed in algebraic geometry is to take a category of local models, the affine schemes, and then glue them together. Now, affine schemes can be as nasty as arbitrary schemes. They are arbitrary algebraically defined closed subsets of nn-dimensional (complex, say) space. They can be very singular, nn can be infinite, etc. But the equivalence relations you use to glue them together are *very* nice. The equivalence relations E⊆X×XE\subseteq X\times X you use have the property that each projection E→XE\to X is required to be etale, which is just the algebraic version of a local isomorphism.

It almost looks to me that your generalization procedure is in some sense orthogonal to this. You take very nice spaces (convex sets or open subsets of RnR^n), but then you glue them together along any equivalence relation with the property that the quotient is concrete. Is that right? So if we remember that any sheaf is a colimit of representable ones, then the key difference between your categories CC and the whole topos is in some sense measured by the failure of an arbitrary sheaf to be concrete. Is that right?

2. Suppose XX is a space in one of your categories CC, and E⊆X×XE\subseteq X\times X is an equivalence relation, where EE is also in CC. Are there nice sufficient conditions on EE guaranteeing that the quotient X/EX/E in the ambient topos actually lies in CC?

3. Is it true that if you glue two lines together at one point (using the obvious colimit), the space you get will be isomorphic to the zero set of the function f(x,y)=xyf(x,y)=x y in the plane? I’m going to guess it’s not. (For comparison, this is true in the category of algebraic spaces, as long as you glue the lines transversely, but it isn’t true in the ambient topos, the category of sheaves of sets on the category of affine schemes with the etale topology.)

Re: Convenient Categories of Smooth Spaces

James wrote:

You take very nice spaces (convex sets or open subsets of ℝn\mathbb{R}^n), but then you glue them together along any equivalence relation with the property that the quotient is concrete. Is that right?

Of course this isn’t how we define smooth spaces. And, it’s far from how I usually think of them. So, let me struggle to take my usual way of thinking and express it in a way that might help answer this question.

A smooth space is a concrete sheaf on a certain site. Your ‘very nice’ smooth spaces are the ‘representable presheaves’ on this site, which are specially nice concrete sheaves. Every smooth space is a colimit of these representables.

In some rough sense, we can form this colimit by taking a big fat disjoint union of ‘very nice’ smooth spaces and then modding out by an equivalence relation.

But, when I say ‘disjoint union’ and ‘quotient’, I secretly mean ‘coproduct’ and ‘coequalizer’. And these are just a long-winded way of saying ‘colimit’. And we need to know which category we’re in when we take this colimit. There are lots of choice! We’ve got the category of concrete sheaves (= smooth spaces in our application), the category of sheaves, the category of concrete presheaves, the category of presheaves, and the category of sets — since every concrete presheaf or sheaf has an underlying set! These categories are related by a charming network of functors.

I’m not 100 percent sure which category you’re talking about in your penultimate sentence above. Maybe you mean “you take the colimit in the category of sheaves, but make sure this colimit has some property that ensures the resulting sheaf is concrete”?

I don’t think of it this way, though quite possibly one could and should. Instead, I think of it one of two other ways: naive and less naive.

The naive way relies heavily on the fact that any concrete sheaf (= smooth space, in our application) has an underlying set. The ‘underlying set’ functor is faithful, so a concrete sheaf is just a set with extra structure: a ‘smooth structure’. In this way of thinking, I can form any smooth space by taking a bunch of smooth spaces, taking the disjoint union of their underlying sets, modding out by any equivalence relation, and then equipping the resulting set with a suitable smooth structure.

In the less naive way of thinking, I form any concrete sheaf as a colimit of representables. This colimit is taken in the category of concrete sheaves. How do I take this colimit? First I take the colimit of the underlying presheaves (which is easy). The resulting presheaf may not be concrete — but I then force it to be concrete using a functor cleverly named ‘concretization’. This is a left adjoint so it preserves colimits. I have now obtained the colimit in the category of concrete presheaves. But, the resulting concrete presheaf may not be a sheaf. So, I force it to be a sheaf using a functor cleverly named ‘sheafification’. This is a left adjoint so it preserves colimits. It also preserves the property of concreteness. I have now obtained the colimit in the category of concrete sheaves!

Somehow the abstract baloney in the previous paragraph is really just a painful explanation of the word ‘suitable’ in the paragraph before that.

Re: Convenient Categories of Smooth Spaces

Another small typo: on p.23, in the last paragraph before section 5, the description of going from an ‘FF-space’ to an abstract simplicial complexes doesn’t seem quite right.

Given an a.s.c. (X,K)(X,K), the associated FF-space should have X(n)={f:n→X|im(f)∈K}X(n) = \{ f: n \rightarrow X | im(f) \in K \} (not just {\{sets in KK of size n}n\}), I think—essentially, since F(m,n)F(m,n) includes all maps m→nm \rightarrow n, not just monos, we have to throw degenerate simplices into X(n)X(n).

Re: Convenient Categories of Smooth Spaces

Thanks! I guess you’re right: if we want to treat an abstract simplicial complex as a special sort of presheaf on FF (the category of finite sets and functions), it’ll need to have degeneracies as well as face maps.

Re: Convenient Categories of Smooth Spaces

Hi John and Alex,

I’ve now had time to read through your paper in some detail. Obviously, my main interest is in where it directly relates to my paper (though all of it is interesting) but as I read I wielded my dread red pen and so you may as well reap the benefit.

Let me make it absolutely clear from the outset that I think it is a good paper and easy to read. My comments, particularly the nit-picky ones, are not to be taken as criticisms of the paper overall. Rather, as I read it then whenever I thought “I don’t immediately see what you’re getting at there”, then I marked it. The key is the ‘immediately’! It usually was the case that I figured it out after a moment or two, but for the majority of the paper then I didn’t have to do that so when I did have to pause then it was worth marking. Also, most of my comments are ‘judgement calls’ and I’m being at my most nit-picky because it’s your job to write the paper and if I don’t tell you something that seemed a little odd to me then you don’t have a choice as to whether or not to change it.

Some general comments first.

In sections 3 and 5 you do a lot on categorical properties of your categories – section 3 descriptive and section 5 provides the proofs. In the abstract you make it clear that all the category stuff is there for “differential geometers” who, by implication, do not know much category theory. It therefore seems a little odd that you never explicitly say:

We do all these constructions by doing the construction to the underlying set and then “imposing” a suitable smooth structure on it.

In section 3 this would just make a little clearer what you are actually doing. However, in section 5 this would simplify things a lot, I think. Basically, “concrete sheaves on a concrete site” is an example of a “topological category over Set” in which case all of these “standard” constructions are formed by doing them for the underlying sets and then imposing a suitable smooth structure.

I think this approach would save a lot of time and make the categorical stuff easier to read for non-categorists (after all, anyone familiar enough with basic category theory is going to skip most of section 5. The only part of that section that I really paid attention for was 5.2 – mainly since I have yet to answer Todd’s question on local cartesian closedness of Frölicher spaces posed over here and I was looking for inspiration.) Where it would also save a fair bit of effort is with colimits. After all, it is fairly intuitive to see that one can make colimits by forming the colimit of the underlying set and then imposing a suitable smooth structure. On the other hand, the concretisation of sheaffification (how many ‘f’s in that?) is obscure in the least!

Section 5 is also a little oddly balanced in that you are very detailed in the early sections: Subspaces and Mapping Spaces, but very abstract in the Colimit section.

The inclusion of simplicial complexes is a little strange. It’s main role, it seems, is to provide a further example of a category of “concrete sheaves over a concrete site”. Certainly it appears to have little to do with categories of “Smooth Spaces”. However, this is never actually said and, moreover, it is given quite high prominence in the early stages – even getting a mention in the abstract.

In relation to my paper, there are a few hang-overs from my mistaken assumption that the convex sets be closed. In particular, the “locally smoothly extendible” functor has changed. Full details will appear in the next draft, but the basic idea is to factor the plots of the Chen space through plots of the original Souriau space. If the convex sets were closed, Whitney’s theorem would give the “locally extendible” description but without the closed assumption, we can no longer describe them in such a fashion. The problem with “locally extendible” (which, after all, makes sense for any choice of subset of ℝn\mathbb{R}^n) is that the composition of a locally extendible map with a smooth map need not be locally extendible again.

Now some slightly more focussed comments.

Kriegl and Michor’s work. The references to their work are not quite as they should be. In the introduction you imply that they construct a cartesian closed category of manifolds. In fact, a careful read of the introduction of Chapter IX of their book reveals that this is precisely what they don’t do. They have done it, they say, elsewhere but

Unfortunately they [Kriegl and Michor] found no applications, and even the authors [Kriegl and Michor again] were not courageous enough to pursue them further and to include them in this book.

What they do do in their book is find a cartesian closed category of smooth linear spaces. The best non-linear ambient cartesian closed category would be that of Frölicher spaces.

Incidentally, at the end of the first part of section 2 you mention their Theorem 24.5 as part of why life is simpler when the domains are closed. Their theorem makes no assumption on closedness and so applies to all convex sets. This is probably another symptom of my error in thinking that Chen’s domains were closed. What makes things easier in the closed setting is Whitney’s theorem on extensions of smooth maps.

Incidentally, I don’t know whether Kriegl and Michor are the originators of this theorem or not. They do not attribute it to anyone else in their book, but they may view it as minor enough not to require attribution.

Mostow’s contribution. Mostow did not invent his version of “differentiable space”, although he does not make this clear when he gives the definition. On p266 he writes

The definition which we have given of differentiable space appeared in the work of R. Sikorski [29], but he did not define differential forms in the way we did.

Incidentally, I have not come across any definition of “differentiable space” which used coplots and which had target spaces other than ℝ\mathbb{R} for those coplots.

Wherefore art thou concrete? After definition 15 you say

Finally, axiom 3 implies that any Chen space X gives a special kind of sheaf on the site Chen: a ‘concrete’ sheaf, …

‘Fraid not. Axiom 3 just says that the ‘underlying set’ functor agrees with the ‘set of points’ functor; in other words, the plots ‘see’ the whole of the set. It says nothing about concreteness. The concreteness is sort-of built in to the definition so is more of an underlying assumption. I don’t think you can separate it out into an axiom without making the whole definition much more abstract.

Terminal objects. Is there a reason why the definition of a concrete site requires the terminal object? Couldn’t I just take any separator? I want it to be a terminal object for looking at equivalences of categories of smooth objects, but I’m not sure I want it to be an assumption. For example, if I had an algebraic theory then the underlying set functor is represented by the free gadget on one element whereas the terminal object has underlying set the terminal object. These are rarely the same thing. So what if my site were built out of algebras in some fashion? Could I still have concrete sheaves over it?

Finally, some even more picky points.

Introduction.

Abstract. Bit on the waffly side, I thought. If I were just scanning this on the arXiv I’d’ve probably missed the main point: nice objects versus nice categories.

Axiom 3 of diffeological space: points in ℝn\mathbb{R}^n are not usually (!) open.

On Chen spaces: “Here we use his final, most refined approach”. In retrospect, was it really the most refined? That’s a little debatable since his first forcing condition was actually stronger than the sheaf condition.

On Chen’s evolution. Maybe emphasise the changes a little more, particularly where the topology is concerned. For example, the 1975 definition did indeed drop the Hausdorff condition but it still retained the topology.

“Pythagorean triples” is this really relevant?

Paragraph “Situations like this”. You are using the fact that a map into a convex region is smooth if and only if it is smooth into its ambient vector space. As you are going into quite some detail about smoothness between convex sets, this is quite an important point to make. It explains why all the focus is on the domain and not the codomain.

2.1 Examples. I get a little lost in distinguishing between “smooth” in the usual sense and “smooth” as a morphism between Chen (or diffeological) spaces.

“with better formal properties”. Would “categorical” be better here? Also, much of the rest of the paper is to justify this sentence so stating it here as a fact downplays the rest of the paper a little.

With regard to the functor C∞→TopC^\infty \to Top, there is another natural functor given by taking the projective topology for all smooth functions on the Chen space (i.e. morphisms to ℝ\mathbb{R} with its standard Chen structure). I think that these provide left and right adjunctions to the functor Top→C∞Top \to C^\infty which assigns to a topological space all its continuous plots.

2.2 Comparison

“identity on maps” issue, I guess the point is that the underlying set functor is faithful and so both So(f)So(f) and ff have the same underlying set map.

Last line of paragraph beginning “The embedding Ch♯Ch^\sharp lacks this defect”. The Ch♯Ch^\sharp is typeset wrong (italic instead of roman).

Convenient Properties.

Is “strong” the more usual terminology? I’ve been using “extremal” but I’m newish to this game so I don’t know.

Smooth Spaces as Generalised Spaces

“Axiom 1 in the definition of a Chen space”

Definition 16. The displayed maths in this definition should really be taking place in SetSet so there should be lots of hom(1,?)hom(1,?)s in it.

Definition 18. Any chance of modifying the “D space” notation a little? I don’t like the actual space (rather than the word “space”!) in it. Putting in a hyphen or something would make it clearer that I should read “D-space” as one thing.

Lemma 21. A Cj→CC_j \to C has snuck in here instead of Uj→UU_j \to U.

Proposition 25. Err… check axiom 1? (Sorry, couldn’t resist).

Convenient Properties

Proposition 27. In the proof, the “underlying set” functor didn’t typeset so well (bit on the short side).

Proposition 28. Careful of circular arguments (or being seen to be circular). It would be clearer if, in the proof, you make it clear that the Ω\Omega that you start with is the functor D↦2D̲D \mapsto 2^{\underline{D}} and that you will go on to show that it is a concrete sheaf.

Definition 31. “We say a morphism of”

Generally, in this section, the typesetting of your diagrams goes a little skew-if when the arrows have tails or strange heads. Are you using xy.sty? If so, you can add a little spacing here and there to sort this out. I don’t know about other packages.

After Definition 41. “We now prepare to show”. Yuk! I know you guys are American, but really! You’re not related to Humphrey Appleby are you?

Grothendieck plus construction. Is there a smallness assumption needed here? Perhaps this is implicit somewhere.

Chen’s predifferentiable spaces. It’s a bit strong to say that he “described a systematic process”. He gives a one line description of the subsequent plots. On the other hand, his one line description is a lot easier to “see” than the plus construction.

Well, I think that’s enough to be going on with! I ought to get back to improving Comparative Smootheology. Feel free to rip that apart in revenge!

Re: Convenient Categories of Smooth Spaces

For this very reason, I don’t have the energy to respond to your comments here right now, except for a few — not the most important ones, just the most amusing:

The inclusion of simplicial complexes is a little strange.

It’s a bit of a digression, but this example gives some interesting results: for example, Todd Trimble was surprised to hear that the category of simplicial complexes is locally cartesian closed. It’s also one of the simplest interesting examples of a category of concrete sheaves.

Is there a reason why the definition of a concrete site requires the terminal object? Couldn’t I just take any separator?

Perhaps — indeed Dubuc, whose work we cite, considers extremely general ‘underlying’ functors, where the underlying category doesn’t even need to be the category of sets. But, we had a lot of trouble following the details of his work. So, we decided to go to the opposite extreme, and keep the formalism as un-general as possible — just general enough to handle our examples.

Is “strong” the more usual terminology?
I’ve been using “extremal” but I’m newish to this game so I don’t know.

Any chance of modifying the “D space” notation a little? I don’t like the actual space (rather than the word “space”!) in it. Putting in a hyphen or something would make it clearer that I should read “D-space” as one thing.

Oh, but that would ruin the best joke of the paper: the way that ‘D Space’ becomes ‘Chen space’ when we set D = Chen. Nothing should ever stand in the way of a joke.

Re: Convenient Categories of Smooth Spaces

After reading this paper, I pondered Chen’s different definitions again and realised that his middle definition is a sheaf condition except that the site is not what one would expect. I haven’t checked the details (i.e. that this does genuinely give a site). The idea is to take the category of convex subsets (closed if you like) with C∞C^\infty maps as morphisms and declare a family of maps into an object, say CC, to be a covering if it is deterministic for smooth maps. That is, a map f:C→C′f : C \to C' is smooth if and only if fθf \theta is smooth for all θ:C″→C\theta : C'' \to C in the family. This ought to define a site. Then the Chen spaces in the sense of the 1975 definition are the quasi-representable (that is, concrete) sheaves on this site.

So Urs’ comment somewhere else was exactly wrong. It wasn’t that Chen was gradually converging on the idea of a sheaf. The concept of a sheaf is there as soon as he realised the need for some sort of forcing condition. Rather, he was unsure as to the correct site. Thus in going from the 1975 to 1977 definitions, he only changed the site – albeit in two ways.

This doesn’t, of course, mean that Chen knew that the original definition was sheafy.

I don’t know whether or not it’s worth pursuing the idea of the “smooth site” any further (for example, to check that it is actually a site). But it’s certainly worth noting that there is an intermediate step in between “q-r sheaves on the usual site of convex sets” and Frölicher spaces which will generalise to a non-set-based theory.

These then lie in between “very nice spaces” and “extremely nice spaces”. “Pretty nice spaces”?

Re: Convenient Categories of Smooth Spaces

Andrew wrote:

After reading this paper, I pondered Chen’s different definitions again and realised that his middle definition is a sheaf condition except that the site is not what one would expect.

Right. I noticed this when reading your earlier discussion with Urs, but didn’t point it out, out of a (doubtless misguided) desire not to draw more attention to these sheafy ideas before Alex’s work appeared in our paper!

(I’m sort of protective of my students.)

Yes, there should be a very general way to impose a Grothendieck topology on a category by saying that a family fi:Xi→Xf_i : X_i \to X is a covering iff it is jointly epimorphic, i.e. iff two morphisms g,h:X→Yg,h: X \to Y must be equal when their composites with all morphisms fif_i agree. There may be some conditions required on the category for this to give a Grothendieck topology… I haven’t checked, but experts would already know.

Smooth Site

I’ve thought a bit more about this and I no longer believe it. I’m not an expert in sites, despite having just read the Wikipedia article on Grothendieck topology, so I’m using topological spaces as a guide for my intuition. Experts may wish to jump in and correct my inanities (where’s Todd when you need him?).

The problem is that the maps between “coverings” go the wrong way. For a topological space, when we have a cover 𝒰\mathcal{U} of XX and a morphism f:Y→Xf : Y \to X then there is a cover 𝒱\mathcal{V} of YY such that for each V∈𝒱V \in \mathcal{V}, there is a U∈𝒰U \in \mathcal{U} and a morphism g:U→Vg : U \to V making the following diagram commute.

However, if we take the category of (closed if you like) convex subsets of Euclidean spaces with C∞C^\infty maps as morphisms, and define a covering to be as Chen had it in 1975, namely a family 𝒰\mathcal{U} covers XX if for every set (or maybe continuous) map f:X→X′f : X \to X', ff is a C∞C^\infty map if and only if fcf c is C∞C^\infty for all c∈𝒰c \in \mathcal{U}. Then for the correct transformation, we consider a smooth map f:Y→Xf : Y \to X. Given a morphism c:U→Xc : U \to X which appears in some cover of XX we want to find a suitable morphism d:V→Yd : V \to Y and a morphism g:V→Ug : V \to U making the diagram commute as above.

Unfortunately, this isn’t always possible. Take XX to be the figure 8 and YY to be the circle with the map Y→XY \to X identifying, say, (1,0)(1,0) and (−1,0)(-1,0). By Boman’s result, we know that the family of smooth curves forms a “covering”. So here’s a smooth curve in XX: map (−∞,0)(-\infty,0) smoothly onto the arc from (0,1)(0,1) to (1,0)(1,0) bijectively and so that as we approach 00 then we approach (1,0)(1,0) infinitely slowly. Similarly, map (0,∞)(0,\infty) onto the arc from (0,−1)(0,-1) to (−1,0)(-1,0). Then we map 00 to the crossing point, (1,0)=(−1,0)(1,0) = (-1,0). This is a smooth curve from ℝ\mathbb{R} to YY. This doesn’t lift to the circle since you’d have to break ℝ\mathbb{R} at 00 and this isn’t allowed for convex sets.

It may be possible to make this work in a larger category, perhaps Chen spaces or Frölicher spaces, but that sort of defeats the purpose.

By the way, your example wherein you want to consider jointly epimorphic families of morphisms isn’t a generalisation of what I was suggesting. Chen’s 1975 condition says that we take families that distinguish C∞C^\infty morphisms from non-C∞C^\infty morphisms. What you are suggesting would be a family that would distinguish one C∞C^\infty morphism from anotherC∞C^\infty morphism. Close, but no cigar (or coffee).

Still, it feels as though there ought to be some general context for this kind of structure. Anyone know?

Re: Convenient Categories of Smooth Spaces

Andrew wrote:

Chen’s 1975 condition says that we take families that distinguish C∞C^\infty morphisms from non-C∞C^\infty morphisms. What you are suggesting would be a family that would distinguish one C∞C^\infty morphism from another C∞C^\infty morphism.

Oh, okay. I guess I’ve never thought enough about his 1975 definition.

Re: Convenient Categories of Smooth Spaces

I guess it’s the time-zone that means that I’m the first to notice this paper going on the arXiv. Great!

I’ll read it properly and thoroughly when I next get close to a printer.

One thing I pondered on a casual look-through was your Cantor set example of how bad Chen spaces can be. Actually, I’m not sure that this is a particularly bad space. Presumably we give it the smooth structure as a subset of the real line, whereupon its plots are those plots of the real line which factor through the Cantor set. Seems to me as if all of these should be constant, and thus the Cantor set gets a discrete smooth structure which, apart from its sheer size, is not particularly scary.

Am I missing something here?

Added in proof: The “Great!” in the first paragraph was linked to the appearance of the paper on the arXiv, not to the fact that I was the first to comment on it.

Re: Convenient Categories of Smooth Spaces

Yeah, okay, I just want the glory and kudos.

Seriously, I’ve had a read through and have a few comments. Only a few, you’ll be pleased to hear! Some of these I’ve said before so presumably you’ve thought about them; let me know which I should shut up about!

Introduction

I think you’re still misrepresenting Kriegl and Michor’s work slightly (here and in section 2). The “convenient” in their book title refers to the linear category. I gave a quote in my earlier comment about the non-linear case. Also, their theorem on convex sets does not assume that those sets are closed, whereas your quote in section 2 at the least implies that it is needed.

Paragraph beginning “In 1977”, phrase “since every smooth map from a convex subset of anyℝn\mathbb{R}^n should count as a plot.”. This is somewhat confusing; I presume that you mean the target of these maps to be some fixed smooth manifold rather than an arbitrary Chen space (wherein it is tautologous).

Paragraph beginning “In 1980”, first sentence. The word “convenient” is extremely loaded in this context! Did Souriau show at the outset that this category has all the properties that this word implies? Isn’t part of the point of your paper showing that this is so?

Same paragraph, sentence “Each approach has its own advantages … manifolds with corners are more easily studied as Chen spaces.”. I find myself naturally asking “Why?”. Surely this is the case only if one thinks about what the manifold looks like locally, but isn’t that precisely what the plots of Chen and diffeological spaces are not there for! I can define the notion of “local picture” equally well in Chen and Souriau for manifolds-with-corners and study them with equal aplomb. I can say that a manifold-with-corners looks locally like the diffeological spaceℝ+n\mathbb{R}^n_+, where this is given its standard diffeological structure. I worry that this sentence (without further amplification) falls into exactly the trap you warned us about in the previous paragraph.

Paragraph beginning: “All this convenience comes with a price: ..”. I’m still a bit disappointed with the Cantor set example. Isn’t there a more exciting example ready to hand? I think that quotients might be more exotic.

Paragraph beginning: “Most other approaches …”. For “maps out” I haven’t come across any that doesn’t use only ℝ\mathbb{R} for the targets. Also, the final sentence (“Our work is designed …”), it doesn’t work for all “maps in” approaches but only those with the sheaf property. For example, it wouldn’t (necessarily) work with Chen’s other definitions.

Smooth Spaces

Paragraph “Chen spaces are a bit subtler”. Really? I thought that the point was that formally, Chen spaces and diffeological spaces were very similar. The subtlety, such that it is, comes in in knowing what a smooth map between convex sets is and this comes later. The following definition does not reveal any of the subtlety in the definition so one is left wondering whether it was so subtle that one missed it.

(By the way, do I get a prize for a sentence with “in in” in it?)

I’ve already mentioned the Kriegl and Michor issue in this section.

Near the end (of this section), paragraph “The embedding Ch♯Ch^\sharp …”, the last Ch♯Ch^\sharp is not typeset correctly.

Finally, thinking particularly about the stuff on colimits but also other things, you seem to think of Chen spaces as “special sheaves” rather than “sets with structure”. However, I get the impression that you are trying to sell this as “generalised manifolds” which more fits the “sets with structure” than “special sheaves”. Introducing sheaffification and concretisation purely to prove that these categories have colimits seems overkill - can’t one just construct the colimit directly by taking the colimit of the underlying sets and then putting the obvious structure on it? To me, this seems far far simpler than dealing with sheaves.

I guess I’m harking back to my comment that a lot of the structure can be subsumed in the statement “D Spaces is topological over Set”. On the one hand, that’s just a box to be unpacked and it has the same contents as you already have. However, it has the advantage of saying “This category that I’m telling you about is quite a bit like that category over there that you already know a lot about.”. Then you go on to say what the differences are (cartesian and locally cartesian closed).

Re: Convenient Categories of Smooth Spaces

Re: Convenient Categories of Smooth Spaces

Thanks for the new round of comments, Andrew. I hope that when we submit this paper for publication — where? does anyone have suggestions? — you are the referee. And I hope that when you referee it, you write: “Thanks to the careful vetting this paper has received at the n-Category Café, it needs no further improvements.”

Just a couple of comments on your comments, for now:

Should this be DSpaceDSpace? If not, what’s DSetDSet?

There’s nothing called DSetDSet in our paper; we must have meant DSpaceDSpace.

Finally, thinking particularly about the stuff on colimits but also other things, you seem to think of Chen spaces as “special sheaves” rather than “sets with structure”. However, I get the impression that you are trying to sell this as “generalised manifolds” which more fits the “sets with structure” than “special sheaves”. Introducing sheaffification and concretisation purely to prove that these categories have colimits seems overkill - can’t one just construct the colimit directly by taking the colimit of the underlying sets and then putting the obvious structure on it? To me, this seems far far simpler than dealing with sheaves.

First, I should say that we thought very hard about your earlier comment along these lines. We tried to address your question by adding this remark to Section 3:

More generally, we can compute any colimit of smooth spaces
by taking the colimit of the underlying sets and endowing the
result with a suitable smooth structure. For a proof that
C∞C^\infty has all colimits, see Prop. 48. The forgetful functor from C∞C^\infty to SetSet preserves colimits because
it is the left adjoint of the functor equipping any set with its
indiscrete smooth structure.

(I should make it a bit more clear that the last two sentences constitute a proof of the first one!)

Second, we decided long ago that nobody who disliked sheaves would bother reading past section 3, and we designed the paper that way. If someone stops at section 3, they’ll get precise statements of most of our results on smooth spaces, without ever needing to confront sheaves or the main technical idea of our paper: ‘concrete sheaves on a concrete site’. Anyone daring to go beyond that point must learn to love sheaves.

Third, we love sheaves!

We think that for those who can stomach the abstraction, it’s very nice to think of smooth spaces as forming a quasitopos of concrete sheaves on a concrete site, and studying them with the help of the larger topos of all sheaves on this site. (These other ‘nonconcrete’ sheaves include spaces like those studied in synthetic differential geometry: for example, infinitesimal spaces like ‘the walking tangent vector’.)

In particular, it’s very nice to see that the inclusion of the concrete sheaves in the category of all sheaves admits a left adjoint, ‘concretization’… just as the inclusion of the sheaves in the category of presheaves admits a well-known left adjoint, ‘sheafification’. Since left adjoints preserve colimits, this makes it fairly evident that we can take the colimit of concrete sheaves by taking the colimit of their underlying presheaves, then concretizing, and then sheafifying.

We were unable to resist the charm of this argument. We actually tried, thanks to your previous comments! But we soon
realized that the ‘suitable’ smooth structure on the colimit of the underlying sets — which you’re calling the ‘obvious’ smooth structure — is exactly the smooth structure you get from sheafifying. In other words, roughly speaking, a plot in a colimit of smooth spaces is formed by gluing together plots in these spaces. This is what sheafification does!

(Or more precisely, this is what Grothendieck’s plus construction does. In general you need to apply this construction twice to turn a presheaf into a sheaf. In our situation, one turn of the crank is enough.)

Given this, and given that we’d already introduced the technology of sheaves on sites to prove all our other theorems,
it seemed tiresome to describe the process of constructing plots on the colimit without admitting its true name: sheafification.

Of course, I can imagine a better version of our paper where we give a less strenuous approach to colimits before explaining that it is in fact sheafification. But life is short, and there are many papers left to write…

…many miles to go before I sleep…

Hmm, maybe it would be easy to include a few remarks in the above-quoted paragraph, explaining in simple terms how to get the smooth structure on a colimit of smooth spaces, and saying that this is the same thing as ‘sheafification’. That might be helpful, and not too much work!

Re: Convenient Categories of Smooth Spaces

you seem to think of Chen spaces as “special sheaves” rather than “sets with structure”.

The way to think about it is that a diffeological space XX is a set XsX_s equipped with the extra structure that makes XsX_s the set appearing in a concrete sheaf.

Here’s how I think about it. Given a concrete site, any sheaf on this site has an underlying set: its set of ‘points’. This gives a forgetful functor from sheaves to sets. In general this functor may fail to be faithful. This means that a sheaf can be thought of as its set of points equipped with extra stuff. But we define a notion of ‘concrete sheaf’ on a concrete site, such that the forgetful functor from concrete sheaves to sets is faithful. This means that a concrete sheaf can be thought of as its set of points equipped with extra structure.

Anyone who doesn’t know the precise definitions of ‘properties’, ‘structure’ and ‘stuff’ can learn about them in Section 2.4 here. But my main point is that for a sheaf to be ‘concrete’ is just another way of saying that we can regard it as a set with extra structure. This is what’s special about concrete sheaves! So, there’s indeed no conflict between thinking of them as ‘special sheaves’ and thinking of them as ‘sets with extra structure’.

More interestingly, we can concretize any sheaf on a concrete site, which is a way of forcing the extra stuff to become merely extra structure.

Re: Convenient Categories of Smooth Spaces

More interestingly, we can concretize any sheaf on a concrete site, which is a way of forcing the extra stuff to become merely extra structure.

Hm, how does one do that? Is that in your article?

I thought this was impossible, since there are non-concrete sheaves on concrete sites which have a single point but nontrivial higher curves. The only concrete sheaf with a single point, however, is that represented by the point. It seems to me.

We actually describe a way to concretize any presheaf: to do this, you identify two plots whenever their underlying functions are the same. Lemma 45 shows that the resulting ‘concretization’ functor is left adjoint to the forgetful functor from concrete presheaves to presheaves.

(Warning: I’m using the word ‘plot’ to mean ‘section of a presheaf on a concrete site’, since that’s how people talk in the theory of Chen spaces and diffeological spaces.)

If you have a sheaf and want to make it concrete, you can first concretize its underlying presheaf and then sheafify the result. Lemma 47 assures you that this gives a concrete sheaf.

All these constructions show up when we take a colimit of concrete sheaves! To do this, we first take the colimit of their underlying presheaves, then concretize the result, and then sheafify it.

This is similar to the usual recipe for taking the colimit of sheaves, where you take the colimit of underlying presheaves and then sheafify the result.

I thought this was impossible, since there are non-concrete sheaves on concrete sites which have a single point but nontrivial higher curves. The only concrete sheaf with a single point, however, is that represented by the point. It seems to me.

Indeed: there may be zillions of sheaves whose underlying set has one point, but only one of these is concrete — namely, the terminal sheaf.

Concretization followed by sheafification takes all these zillions of sheaves and turns them all into the terminal sheaf!

So, you can concretize any sheaf, but this can be a highly destructive process.

“forcing the extra stuff to become merely extra structure” – in the meta-system trinity of Stuff, Structure, Properties?

If so, turning some stuff into structure does what (if anything) to the Properties?

In what sense (if any) are Concretization and Sheafification adjoint?

I’m not crazy about the term Concretization anyway. It sounds like a conflation of Reification (making the abstract concrete) and what mobsters do before riverification (speaking of the Seine as analogue of the Hudson).

As to the gargoyles – wasn’t Medusa famous for Concretization of anyone until Perseus applied mirror-symmetry?

Re: Convenient Categories of Smooth Spaces

Ah, so this concretization seems to be closely related (maybe the same? haven’t tried to check) to the construction Andrew described
here, which, the way I read it here, takes a sheaf XX and extracts a sheaf concretely realized on the set of points of XX.

So, you can concretize any sheaf, but this can be a highly destructive process.

So it is not so much that the original sheaf is realized as a concrete sheaf (which the term “concretization” made me think of), but rather that a concrete sheaf is in a way projected out from the sheaf.

Re: Convenient Categories of Smooth Spaces

I think I first mentioned what John and Alex call “concretization” (but which they should, of course, call “concretisation”) here in the first comparative smootheology thread. I also pondered a different process which inflated an arbitrary sheaf to a concrete sheaf, namely by adding in enough extra points. Thus every sheaf becomes trapped between two concrete sheaves - which sounds an incredibly dangerous position to be in!

(By the way; Urs, you shouldn’t complain too strenuously about the word “concretisation”. I can think of lots of things that lose essential properties when put in concrete.)

Re: Convenient Categories of Smooth Spaces

I think of concretization as ‘forcing a presheaf to be concrete’, since it’s left adjoint to the forgetful functor from concrete presheaves to presheaves. As usual, forcing an object to be something it wasn’t can be damaging to its true nature.

For XX a sheaf we get
image(postpt,−,X):Sop→Set
image(post_{pt,-,X})
:
S^{op} \to Set
and
when restricted to representables this yields a presheaf: this is the concretization of XX before re-sheafification I think.

There are two maps that come with the image: the map into it and the map out of it. The former gives the projection morphism of a sheaf on its concretization. The latter gives the injection of the concrete plots into the maps into the set of points.

Re: Convenient Categories of Smooth Spaces

One fun thing about this is that one sees directly that also ptpt can still be varied, so that we can consider “concretization” with values in sheafes that have an underlying presheaf of certain kind.

Sounds ridiculous, but is useful. With Christoph Sachse we are working through examples of integrating super L∞L_\infty-algebras gg by computing the fundamental n-groupoid of the classifying spaces of flat forms with values in gg. When doing that, one needs to let ptpt vary over all superpoints to really get a super nn-group.

So there is a useful generalization of “concretization” which produces not generalized spaces with underlying sets, but generalized spaces with underlying presheaves over superpoints.

Re: Convenient Categories of Smooth Spaces

John éscrit:

I hope that when we submit this paper for publication — where? does anyone have suggestions?

I was going to ask the same question about Comparative Smootheology. Not that the latest version is quite ready yet. I got a bit distracted by infinite dimensional Riemannian geometry (rather, co-geometry).

— you are the referee. And I hope that when you referee it, you write: “Thanks to the careful vetting this paper has received at the n-Category Café, it needs no further improvements.”

I suspect that by participating in this discussion then I’ve removed myself from the list of potential referees. But actually you want a completely different referee who, nonetheless, says exactly the same thing. That way we get extra kudos (by the way, Bruce, I said kudos. Very funny though. You should write to the BBC - they’ve got a page for people like you - see the caption competition in particular.).

Hmm, maybe it would be easy to include a few remarks in the above-quoted paragraph, explaining in simple terms how to get the smooth structure on a colimit of smooth spaces, and saying that this is the same thing as ‘sheafification’. That might be helpful, and not too much work!

That’s pretty much all I would suggest. My point is not that there is a distinction between these two views but just that there are two views. I would say something like the following in the introduction:

These are concrete sheaves on concrete sites. This gives us two ways to think of them: as special sheaves or as sets-with-structure. some waffle about why it’s nice to have each viewpoint.

However, I think I was missing the point that the “obvious” Chen structure needs some extra stuff added in to be a Chen structure and identifying the extra stuff is the sheaffffffiffffffication process. So, actually, I partially withdraw my comment except that to say that someone more used to ‘sets-with-structure’ than sheaves may still be interested enough to read the whole paper and may appreciate a few signposts. You may feel that this set is null, and as I now appreciate the issue more fully than I did then possibly the only person in this set has now removed himself from it so it may well be empty. Tant pis!

Re: Convenient Categories of Smooth Spaces

Mind you, I sympathize with Andrew when it come to sheafification. Yesterday I was walking along the Seine and I discovered a set of gargoyles modelled after students who attended Grothendieck’s seminars on algebraic geometry. Different gargoyles illustrate their reactions to different concepts. Here’s one learning about sheafification:

Convenient models of String(G)

Some people are trying to finite finite dimensional models of the String group String(G)String(G) associated to a simple compact and simply connected Lie group GG.

In terms of non-concrete diffeological spaces one can consider the following, which comes pretty close:

recall that, as a diffeological space, the group GG itself is defined as follows:

it is given by the rule which to each test domain UU assigns the set of plots
U↦C∞(U,G)
U \mapsto C^\infty(U,G)
of smooth maps from UU into GG.

Now modify this rule slightly. Let μ3∈Ω3(G)\mu_3 \in \Omega^3(G) be the image in deRham cohomology of a generator of H3(G,ℤ)H^3(G,\mathbb{Z}) and then decree that a plot on test domain UU now is a pair
(f,B)
(f,B)
consisting of a smooth map f:U→Gf : U \to G, as before, but now equipped with a 2-form B∈Ω2(U)B \in \Omega^2(U) such that
f*μ3=dB.
f^* \mu_3 = d B
\,.

Call the non-concrete smooth space defined this way GμG_\mu.

Then

- plots ℝ0→Gμ\mathbb{R}^0 \to G_\mu are the same as plots ℝ0→G\mathbb{R}^0 \to G

- plots ℝ1→Gμ\mathbb{R}^1 \to G_\mu are the same as plots ℝ1→G\mathbb{R}^1 \to G

- plots ℝ2→Gμ\mathbb{R}^2 \to G_\mu are pairs consisting of plots ℝ2→G\mathbb{R}^2 \to G together with a 2-form on ℝ2\mathbb{R}^2.

What is the fiber of this? Over the neutral element of GG (as well as over every other element) the fiber is the smooth space whose plots on UU are closed 2-forms on UU. I called this space S(CE(bu(1)))S(CE(b u(1))). It is a rational approximation to a K(ℤ,2)K(\mathbb{Z},2)

This product is supposed to cover the product on GG. That means that the assignment has to be
((f1,B1),(f2,B2))↦(f1⋅f2,something).
((f_1,B_1),(f_2,B_2))
\mapsto
(f_1 \cdot f_2, something)
\,.

we need to figure out the something on the right.

To do so, we use the fact that the alternating difference
δμ3=d0*μ3+d2*μ3−d1*μ3∈Ω3(G×G)
\delta \mu_3
=
d_0^* \mu_3 + d_2^* \mu_3 - d_1^* \mu_3
\in \Omega^3(G \times G)
of the pullback of μ3\mu_3 along the three maps
di:G×G→G
d_i
:
G \times G \to G
given by projection onto the first factor, projection onto the second factor, and multiplication in GG, is exact: there is ν∈Ω2(G×G)\nu \in \Omega^2(G \times G) such that
δμ3=dν.
\delta \mu_3 = d \nu
\,.
This ν\nu itself has the property that δν=0\delta \nu = 0 in Ω2(G×G×G)\Omega^2(G \times G \times G). In other words (ν,μ)(\nu,\mu) represents a 4-cocycle as simplicial forms on BGB G(d+δ)(ν,μ)=0.
(d+\delta)(\nu,\mu) = 0
\,.

Finally, one needs to check that the associative product we have so far is invertible. I claim that the inverse operation acting on the plot (f,B)(f,B) produces the plot (f−1,−B)(f^{-1},-B). This follows because actually
ν=⟨θ1,θ¯2⟩ \nu = \langle \theta_1,\bar\theta_2 \rangle, with ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle the invariant bilinear form on Lie(G)Lie(G), θ1\theta_1 the left-invariant Maurer-Cartan form on GG and θ2\theta_2 the right-invariant one.

Re: Convenient models of String(G)

The “pretty close” was mainly intended to disclaim that S(CE(bu(1)))S(CE(b u(1))) is not exactly a K(ℤ,2)K(\mathbb{Z},2), but just a “rational model” in some sense.

Your message found me in the middle of writing a private email with more background information: I think the above model arises as follows:

GμG_\mu should be the spatial realization of the smooth fundamental 2-group of the classifying space of flat string(g)string(g)-valued forms.

In symbols:

Gμ=|String(G)|
G_\mu = |String(G)|
where
BString(G):=Π2(S(CE)(gμ3)).
\mathbf{B}String(G)
:=
\Pi_2(S(CE)(g_{\mu_3}))
\,.
Here by “spatial realization” |⋅||\cdot| I mean the operation which sends any smooth ω\omega-category CC to the sheaf
|C|:U↦Hom(Πω(U),C).
|C| : U \mapsto Hom(\Pi_\omega(U),C)
\,.
This is in general different from “geometric realization of the ω\omega-nerve”, but somehow it is also related.

As I said, I think I should be able to prove Gμ=|String(G)|G_\mu = |String(G)| in this sense, but right now I still get a headache when trying to do that.

Read the post Spivak on Derived ManifoldsWeblog: The n-Category CaféExcerpt: David Spivak has an interesting thesis on 'derived differential geometry'.Tracked: August 19, 2008 7:16 PM

Read the post Comparative Smootheology, IIIWeblog: The n-Category CaféExcerpt: The third episode in our continuing comparison of various frameworks for differential geometry.Tracked: September 3, 2008 8:36 PM

Read the post Smooth Differential Graded Algebra?Weblog: The n-Category CaféExcerpt: What is the right definition of generalized smooth differential graded-commutative algebras?Tracked: September 9, 2008 11:04 AM

Re: Convenient Categories of Smooth Spaces

Hi there,

I followed your discussion on diffeology and Chen spaces. I didn’t participate because I had nothing pertinent, related to your questions, to tell. But I must tell you that I have been very interested in reading it. It made me think on a few questions, one of them got a partial answer in my last preprint “Diffeology of manifolds with boundary” (I had a short email exchange about this question with John Baez). I am more interested now, finishing my book, in finding examples and applications of diffeology than developing other aspects of the theory. It seems to me that the philosophy and the methods of diffeology are clear enough now for everybody to understand how to do if he is interested in applying diffeology to his concrete questions. But my curiosity has been excited by what you call String(G) (as a possible new example). Can you explain me what it is about ? Sorry if I am too lazy to look for the definition (and try to google String(G) :-) )

Re: Convenient Categories of Smooth Spaces

But my curiosity has been excited by what you call String(G)String(G) (as a possible new example). Can you explain me what it is about?

There are by now various “models” for String(G)String(G).

One is this:

String(G)String(G) is a crossed complex of Fréchet-Lie groups
Ω^G→PG
\hat \Omega G \to P G
where: GG is a compact, simple and simply connected Lie group, Ω^G\hat \Omega G is the universal central extension of the based loop group of GG and PGP G is the group of based paths in GG.

This is usually modeled as Fréchet manifolds, but in some applications it seems much more natural to consider it instead in terms of diffeological groups.

I was wondering once how the representation theory of loop groups would change were one to replace the Fréchet by the diffeological model. I think Andrew Stacey told me that it is known that the representation theory works the same way in both cases! I have been wanting to follow up on this aspect but have not found the time so far.

Above I was talking about a different “model” for String(G)String(G) (only that it won’t be a proper model, I think, but soemthing like a “smooth rational approximation”).

That, however, is not quite a diffeological space but something slightly more general:

namely it is not a concrete sheaf on test domains, but a non-concrete one.

I can show you more on String(G)String(G) in the diffeological context. For the time being I’ll do so by private email instead of here on the blog.

Re: Convenient Categories of Smooth Spaces

Thanks for your answer Urs. We continue this discussion, on String(G), through private mail, OK. I am open to discuss any point on diffeology, especially if it is related to physical examples. Actually we initiated diffeology (20 years ago now :) to give to mathematical physics a good and flexible framework for all these questions in differential geometry involving spaces which are not manifolds but which carry a natural “smooth structure”.

I take the opportunity of this thread to discuss a point which bothered you. Do we need the Chen approach (with closed simplicies as domains of plots) or the diffeology approach with open sets ? I decided long time ago that I prefer the second one because it is closer the way we are used in ordinary differential geometry (domains of charts). And moreover, any half-space or part, closed or nor not, of any numerical domain carries a natural subset diffeology. So there is no need to add something here (it is my point of view). Thanks to the notion of “modeling spaces” (see the book) we can look for diffeological spaces modeled on half spaces, in finite dimension this give exactly the usual manifolds with boundary. So it is not necessary to add half-spaces as domain for plots. Moreover this definition can be naturally extended to infinite dimensional manifolds with boundary easily by considering half-diffeological vector space, but this is another question (I didn’t look into). Another example of modeling is the (effective) orbifolds, we have proved with Yael Karshon and Moshe zadka that a diffeological space modeled on co-finite quotients R^n/G, where G is some finite sugbroup of GL(n), are just ordinary orbifolds, see here. So, my opinion (which is just an opinion) is that considering domains of plots open is just sufficient, and moreover doesn’t introduce not necessary technicalities. Diffeology is made to simplify our approach on these kind of objects not covered by the traditional differential geometry. Moreover, Chen introduced his definition because he was interested more in the homology/cohomology aspect of these spaces, but we have been (and still I am) more interested in the smooth aspects, and homology and cohomology follow naturally.

Re: Convenient Categories of Smooth Spaces

Hi Patrick,

you write:

I take the opportunity of this thread to discuss a point which bothered you.

[…]

Thanks to the notion of “modeling spaces” (see the book) we can look for diffeological spaces modeled on half spaces, in finite dimension this give exactly the usual manifolds with boundary. So it is not necessary to add half-spaces as domain for plots.

Thanks for the link, that should be helpful!

I might add that I was less concerned about this aspect of open versus closed test domains here than, I think, John, who was looking more into an application (QFT as smooth functors on cobordisms with boundaries and corners) where this small technical aspect has relevance.

While I think the question of test domains is a minor one, after a little thinking and a bit of useful discussion with others (crucially: with Todd Trimble) I have come to favor the category of cartesian spaces as test domains, i.e. all plots are ℝn\mathbb{R}^ns, for some nn. So that’s all open balls for practical purposes.

This category of test domains has the enjoyable property that it serves at the same time as a nice category of co-plots, in that there is a useful story to be told about co-presheaves on cartesian spaces as models for “generalized smooth sections and function algebras” (aka “quantities”).

These we “co-probe” with ℝn\mathbb{R}^ns by checking how they change when change their codomains. As indicated in the first few sections of the book by Moerdijk & Reyes, there is a nice story to be told about generalized spaces and generalized quantities modeled on cartesian spaces.

Re: Convenient Categories of Smooth Spaces

Re: Convenient Categories of Smooth Spaces

A little while ago Dan Christensen pointed out a problem with this paper.

In Proposition 29 we claimed that for a concrete site (as defined by us), any representable presheaf is a sheaf. In other words, we claimed that our concrete sites were subcanonical. This turns out not to be true — though it’s true in all the examples we care about.

Since the paper had already been accepted for publication, we needed to fix it quickly, making only small changes. So, we simply added the requirement that our site be subcanonical. The fixed version is now on the arXiv.

For this deeply interested in these issues: it seems we only use the fact that our site is subcanonical in the proof of Proposition 45, which says that the category of concrete sheaves on such a site is locally cartesian closed. It’s possible that one could prove this a different way — we don’t know.